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The brain serotonergic axons (fibers) are quintessential “stochastic” axons in the sense that their individual trajectories are best described as sample paths of a spatial stochastic process. These fibers are present in high densities in virtually all regions of vertebrate brains; more generally, they appear to be an obligatory component of all nervous systems on this planet (from the dominating arthropods to such small phyla as the kinorhynchs). In mammals, serotonergic fibers are nearly unique in their ability to robustly regenerate in the adult brain, and they have been strongly associated with neural plasticity. We have recently developed several experimental approaches to trace individual serotonergic fibers in the mouse brain (Mays et al., 2022). To further advance the theoretical analyses of their stochastic properties (e.g., the increment covariance structure), we developed a convolutional neural network (CNN) that performs high-throughput analysis of experimental data collected with sub-micrometer resolution. In contrast to a recently developed mesoscale platform that can separate large-caliber fiber segments from the background on the whole-brain scale (Friedmann et al., 2020), our microscale model prioritizes the accuracy and continuity of individual fiber trajectories, an essential element in downstream stochastic analyses. In particular, it seamlessly integrates information about the physical properties of serotonergic fibers and high-resolution experimental data to achieve reliable, fully-automated tracing of trajectories in brain regions with different fiber densities. This 3D-spatial information supports our current theoretical frameworks based on step-wise random walks (Janusonis & Detering, 2019) and continuous-time processes (Janusonis et al., 2020). In a complementary approach, we also investigated whether the structure of the serotonergic fibers may provide useful information for machine learning architectures. Specifically, we studied whether dropout, a standard regularization technique in artificial neural networks, can be matched or improved by virtual serotonergic fibers moving through CNN layers (endowed with the Euclidean metric) and triggering spatially correlated dropout events. This research was funded by NSF CRCNS (#1822517 and #2112862), NIMH (#MH117488), and the California NanoSystems Institute. 

Serotonergic axons (fibers) have a ubiquitous distribution in vertebrate brains, where they form meshworks with well-defined, regionally-specific densities. In humans, perturbations of these densities have been associated with abnormal neural processes, including neuropsychiatric conditions. The self-organization of serotonergic meshworks depends on the cumulative behavior of many serotonergic axons, each one of which has a virtually unpredictable trajectory. In order to bridge the high stochasticity at the microscopic level and the regional stability at the mesoscopic level, we are developing tunable hydrogel systems that can support causal modeling of these processes. These same systems can support future restorative efforts in neural tissue because serotonergic axons are nearly unique in their ability to robustly regenerate in the adult brain. In the study, we extended our research in 2D-primary brainstem cultures (Hingorani et al., 2022) to 3D-hydrogels. Tunable hydrogel scaffolds can closely mimic the mechanical and biochemical properties of actual neural tissue in all three dimensions and are therefore qualitatively different from 2D-environments. However, the integration of these scaffolds with highly sensitive neurons poses unique challenges. As the first step in building a hydrogel-based platform for the bioengineering of serotonergic axons, we studied primary brainstem neurons in several commercially available hydrogel platforms. The viability and dynamics of serotonergic somata and neurites were analyzed at different days in vitro with immunocytochemistry and high-resolution confocal microscopy. In addition, live imaging of neuron growth cones was performed, and the observed dynamics was compared to our extensive database of holotomographic (refractive index-based) recordings in 2D-cultures. The progress and key problems will be discussed. This research was funded by NSF CRCNS (#1822517 and #2112862), NIMH (#MH117488), and the California NanoSystems Institute. 

The self-organization of the serotonergic matrix in the brain is a key unsolved problem in neuroscience. This matrix is composed of extremely long axons (fibers) that originate in the brainstem, invade nearly all brain regions, and accumulate in remarkably high densities in many of them. Serotonergic fibers possess a number of intriguing properties, including the ability to robustly regenerate in the adult brain, the strongly stochastic trajectories, and the poorly understood but consistent association with neural plasticity. We developed several experimental methods that can be used to capture the individual trajectories of serotonergic fibers in the mouse brain, including regions with high fiber densities. These data are essential for stochastic modeling efforts that currently utilize two different frameworks (a step-wise random walk based on the von Mises-Fisher directional distribution and the superdiffusive fractional Brownian motion). In one approach, we show that serotonergic fibers can be experimentally isolated by using transgenic mice with the inducible Cre (under the Tph2-promoter), crossed with a Cre reporter line. While the overall labeling intensity falls below that of the best constitutive model in the field (Migliarini et al., 2013), the inducible Cre allows for control over how many fibers are labeled in high-density regions, thus facilitating their semi-automated tracing. A particularly powerful approach is based on the Brainbow toolbox (Cai et al., 2013) which can be used to randomly “color-code” individual axons. We have developed the first implementation of Brainbow-tagging in the serotonergic system (based on intracranial AAV-injections) and demonstrate its potential in downstream stochastic analyses. In particular, we show that some apparent branching points are different fibers crossing at distances below the limit of optical resolution (even in high-power confocal imaging). Finally, we demonstrate the feasibility of imaging single serotonergic fibers with CUBIC-based tissue clearing and high-resolution light-sheet microscopy (with a 20X objective). This experimental toolbox, integrated with stochastic modeling, can advance the current understanding of the dynamics, robustness, and plasticity of the brain serotonergic system. This research was funded by NSF CRCNS (#1822517 and #2112862), NIMH (#MH117488), and the California NanoSystems Institute. 

Recent experimental and theoretical work by our group has shown that the self-organization of the brain serotonergic matrix is strongly driven by the spatiotemporal dynamics of single serotonergic axons (fibers). The trajectories of these axons are often stochastic in character and can be described by step-wise random walks or time-continuous processes (e.g., fractional Brownian motion). The success of these modeling efforts depends on experimental data that can validate the proposed mathematical frameworks and constrain their parameters. In particular, further progress requires reliable experimental tracking of individual serotonergic axons in time and space. Visualizing this dynamic behavior in vivo is currently extremely difficult because of the high axon densities and other resolution limitations. In this study, we used in vitro systems of mouse primary brainstem neurons to examine serotonergic axons with unprecedented spatiotemporal precision. The high-resolution methods included confocal microscopy, STED super-resolution microscopy, and live imaging with holotomography. We demonstrate that the extension of developing serotonergic axons strongly relies on discrete attachments points on other, non-serotonergic neurons. These membrane anchors are remarkably stable but can be stretched into nano-scale tethers that accommodate the axon’s transitions from neuron to neuron, as it advances through neural tissue. We also show that serotonergic axons can be flat (ribbon-like) and produce screw-like rotations along their trajectory, perhaps to accommodate mechanical constraints. We conclude that the stochastic dynamics of serotonergic axons may be conditioned by the stochastic geometry of neural tissue and, consequently, may reflect it. Our current research includes hydrogels to better understand these processes in controlled artificial environments. Since serotonergic axons are nearly unique in their ability to regenerate in the adult mammalian brain and they support neural plasticity, this research not only advances fundamental neuroscience but can also inform efforts to restore injured neural tissue. This research was funded by NSF CRCNS (#1822517 and #2112862), NIMH (#MH117488), and the California NanoSystems Institute. 

Vertebrate brains have a dual structure, composed of ( i ) axons that can be well-captured with graph-theoretical methods and ( ii ) axons that form a dense matrix in which neurons with precise connections operate. A core part of this matrix is formed by axons (fibers) that store and release 5-hydroxytryptamine (5-HT, serotonin), an ancient neurotransmitter that supports neuroplasticity and has profound implications for mental health. The self-organization of the serotonergic matrix is not well understood, despite recent advances in experimental and theoretical approaches. In particular, individual serotonergic axons produce highly stochastic trajectories, fundamental to the construction of regional fiber densities, but further advances in predictive computer simulations require more accurate experimental information. This study examined single serotonergic axons in culture systems (co-cultures and monolayers), by using a set of complementary high-resolution methods: confocal microscopy, holotomography (refractive index-based live imaging), and super-resolution (STED) microscopy. It shows that serotonergic axon walks in neural tissue may strongly reflect the stochastic geometry of this tissue and it also provides new insights into the morphology and branching properties of serotonergic axons. The proposed experimental platform can support next-generation analyses of the serotonergic matrix, including seamless integration with supercomputing approaches. 

Random dropout has become a standard regularization technique in artificial neural networks (ANNs), but it is currently unknown whether an analogous mechanism exists in biological neural networks (BioNNs). If it does, its structure is likely to be optimized by hundreds of millions of years of evolution, which may suggest novel dropout strategies in large-scale ANNs. We propose that the brain serotonergic fibers (axons) meet some of the expected criteria because of their ubiquitous presence, stochastic structure, and ability to grow throughout the individual’s lifespan. Since the trajectories of serotonergic fibers can be modeled as paths of anomalous diffusion processes, in this proof-of-concept study we investigated a dropout algorithm based on the superdiffusive fractional Brownian motion (FBM). The results demonstrate that serotonergic fibers can potentially implement a dropout-like mechanism in brain tissue, supporting neuroplasticity. They also suggest that mathematical theories of the structure and dynamics of serotonergic fibers can contribute to the design of dropout algorithms in ANNs. 

The neighborhood of virtually every brain neuron contains thin, meandering axons that release serotonin (5-HT). These axons, also referred to as serotonergic fibers, are present in all vertebrate species (from fish to mammals) and are an essential component of biological neural networks. In the mammalian brain, they create dense meshworks that are macroscopically described by densities. It is not known how these densities arise from the trajectories of individual fibers, each of which resembles a unique random-walk path. Solving this problem will advance our understanding of the fundamental structure of neural tissue, including its plasticity and regeneration. Our interdisciplinary program investigates the stochastic structure of serotonergic fibers, by employing a range of experimental, computational, and theoretical methods. Transgenic mouse models (e.g., Brainbow) and brainstem cell cultures are used with advanced microscopy (3D-confocal imaging, STED super-resolution microscopy, holotomography) to visualize individual serotonergic fibers and their trajectories. Serotonergic fibers are modeled as paths of a superdiffusive stochastic process, with a focus on fractional Brownian motion (FBM). The formation of regional fiber densities is tested with supercomputer modeling in neuroanatomically accurate 2D- and 3D-brain-like shapes. Within the same framework, we are developing the mathematical theory of the reflected, branching, and spatially heterogenous FBM. 

The immediate neighborhood of virtually every brain neuron contains thin, meandering axons that release serotonin (5-HT). These axons, also referred to as serotonergic fibers, are present in nearly all studied nervous systems (both vertebrate and invertebrate) and appear to be a key component of biological neural networks. In the mammalian brain, they create dense meshworks that are macroscopically described by densities. It is not known how these densities arise from the trajectories of individual fibers, each of which resembles a unique random-walk path. This poses interesting theoretical questions, solving which will advance our understanding of brain plasticity and regeneration. For example, serotonin-associated psychedelics have recently been shown to promote global brain integration in depression [1], and serotonergic fibers are nearly unique in their ability to robustly regenerate in the adult mammalian brain [2]. We have recently introduced a conceptual framework that treats the serotonergic axons as “stochastic axons.” Stochastic axons are different from axons that support point-to-point connectivity (often studied with graph-theoretical methods) and require novel theoretical approaches. We have shown that serotonergic axons can be potentially modeled as paths of fractional Brownian motion (FBM) in the superdiffusive regime (with the Hurst exponent H > 0.5). Our supercomputing simulations demonstrate that particles driven by reflected FBM (rFBM) accumulate at the border enclosing the shape [3]. Likewise, serotonergic fibers tend to accumulate at the border of neural tissue, in addition to their general similarity to simulated FBM paths [4]. This work expands our previous simulations in 2D-brain-like shapes by considering the full 3D-geometry of the brain. This transition is not trivial and cannot be reduced to independent 2D-sections because increments of FBM trajectories exhibit long-range correlation. Supercomputing simulations of rFMB (H > 0.5) were performed in the reconstructed 3D-geometry of a mouse brain at embryonic day 17 (serotonergic fibers are already well developed at this age and begin to invade the cortical plate). The obtained results were compared to the actual distribution of fibers in the mouse brain. In addition, we obtained preliminary results by simulating rFBM with a region-dependent H. This next step in complexity presents challenges (e.g., it can be highly sensitive to mathematical specifications), but it is necessary for the predictive modeling of interior fiber densities in heterogenous brain tissue.